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does an electron have kinetic energy when attached to a proton? if not, what is it transformed into?

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- Thread starter Lolicon
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That is, if we could measure the kinetic energy of all the electrons in the box at every moment, we would get a number that is not zero but is smaller than the total energy of the system.The expected value of kinetic energy is smaller than the total energy of the system.f

- #1

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does an electron have kinetic energy when attached to a proton? if not, what is it transformed into?

- #2

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A proton and electron bound together is called a hydrogen atom. The hydrogen atom has a set of possible energies, each of which is the sum of the kinetic energy and potential energy.does an electron have kinetic energy when attached to a proton? if not, what is it transformed into?

- #3

Mentor

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Yes, it has KE

- #4

There is a nice result called the Virial theorem which says that if the potential energy of interaction between two bound particles goes as ##V(r) = \lambda r^n##, then the potential and kinetic energies of that system are related via ##2 \langle T \rangle = n \langle V \rangle ##. For an electrostatic interaction that potential has ##n=-1##, so ##2\langle T \rangle = - \langle V \rangle##. Equivalently, the total energy is ##E = \langle T \rangle + \langle V \rangle = - \langle T \rangle = \langle V \rangle/2##.

The energy eigenvalues of a hydrogen atom look like$$E = \frac{-13.6 \text{eV}}{n^2}$$For instance at the ground state, ##n=1##, then ##E = -13.6 \text{eV}##, ##\langle V \rangle = -27.2 \text{eV}## and ##\langle T \rangle = 13.6 \text{eV}##. As you go up energy levels, the potential energy and total energy increase, whilst the kinetic energy decreases.

The energy eigenvalues of a hydrogen atom look like$$E = \frac{-13.6 \text{eV}}{n^2}$$For instance at the ground state, ##n=1##, then ##E = -13.6 \text{eV}##, ##\langle V \rangle = -27.2 \text{eV}## and ##\langle T \rangle = 13.6 \text{eV}##. As you go up energy levels, the potential energy and total energy increase, whilst the kinetic energy decreases.

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- #5

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The energy eigenvalues of a hydrogen atom look like$$E = \frac{-13.6 \text{eV}}{n^2}$$For instance at the ground state, ##n=1##, then ##E = -13.6 \text{eV}##, ##V = -27.2 \text{eV}## and ##T = 13.6 \text{eV}##. As you go up energy levels, the potential energy and total energy increase, whilst the kinetic energy decreases.

Those are, of course, the expected values of ##T## and ##V## for a QM system.

- #6

Those are, of course, the expected values of ##T## and ##V## for a QM system.

Thanks, give me two seconds and I'll put the langles and the rangles where they belong

- #7

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To me, "expected value" has the connotation "It has a definite value, and this is what I expect it to be." Of course, that's not what we're talking about here.

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